By elimination of a,b,cfrom the homogeneous equations  x=ab−c,  y=bc−a, z=ca−bwhere a,b,cnot all zero.

a−xb+xc=0→(1)    ya+b−yc=0→(2)za−zb−c=0→(3)Since a, b, c are not all zero, the above system has a non-trivial solution.By eliminating a,b,cfrom (1),(2),(3),we get1-xxy1-yz-z-1=0     ⇒xy+yz+zx=−1